Geometry - Springer · 18 B. Gr¨unbaum Fig. 4. The different types of isogonal hexagons. The...

Discrete Comput Geom 18:13–52 (1997) Discrete & Computational

Geometry© 1997 Springer-Verlag New York Inc.

Isogonal Prismatoids∗

Branko Grunbaum

Department of Mathematics, University of Washington GN-50,Seattle, WA 98195, [email protected]

Abstract. The study of the polyhedra (in Euclidean 3-space) in which faces may beself-intersecting polygons, and distinct faces may intersect in various ways, was quitefashionable about a century ago. The Kepler–Poinsot regular polyhedra, and several of theirgeneralizations, were investigated about that time by Cayley, Wiener, Badoureau, Fedorov,Hess, Pitsch, and others; the accumulated wisdom was presented in Max Br¨uckner’s well-known bookVielecke und Vielflachein 1900. Despite the intrinsic interest of the topic, andits relations to various other disciplines, there have been very few additional investigationsduring the intervening century, except for discussions of uniform polyhedra. In particular,there has been no mention or clarification of the many errors and other shortcomings ofBruckner’s book. One of our aims is to point out the extent of these inadequacies; theyare illustrated by a discussion ofisogonal prismatoids, the investigation of which is ourmain objective. Aprismatoidis a polyhedron having all its vertices in two parallel planes.Familiar examples are prisms and antiprisms. A polyhedronP is isogonalif all its verticesform one transitivity class under isometric symmetries ofP. Although these restrictionsappear very severe, there exist many different kinds of isogonal prismatoids. Some generalconcepts concerning polyhedra with possible self-intersections are presented, and severalclasses of isogonal prismatoids are discussed in some detail.

1. Introduction

According to Webster [18] aprismatoidis “a polyhedron having all of its vertices in twoparallel planes,” which we always take as distinct. Not every text agrees completely withthis definition, but it is precisely the one we need. This is a very wide class of polyhedra—it includes prisms, antiprisms, pyramids, and many other polyhedra. Therefore we restrictattention toisogonal prismatoids, that is, prismatoids in which all vertices are equivalent

∗ This research was supported in part by NSF Grant DMS-9300657.

14 B. Grunbaum

Fig. 1. Examples of Archimedean prisms and antiprisms.

under isometric symmetries of the polyhedron. We note that we understand the isogo-nality restriction in a strong sense: since prismatoids have two distinguished planes, onlysymmetries which map the pair of planes onto itself are considered. Two classes of ex-amples of isogonal prismatoids are known since antiquity: certain prisms and antiprisms.Theprismsandantiprismshave two congruentbases(which are polygons contained inthe parallel planes mentioned in the definition) together with amantleconnecting thetwo bases; the mantle consists of quadrangles in the case of prisms and triangles in thecase of antiprisms. Figure 1 shows examples ofArchimedean prisms, in which the basesare regular convex polygons and the mantle faces are squares, as well as examples ofArchimedean antiprisms, in which the bases are regular convex polygons and the mantleconsists of equilateral triangles. It is well known that every convex isogonal prisma-toid is isomorphic(or combinatorially equivalent) to an Archimedean prism or anti-prism.

The aim of this paper is to investigate more general isogonal prismatoids, in whichfaces may be self-intersecting polygons, and different faces may have intersections thatare not common edges or vertices. As we shall see, this enlarges the family of isogonalprismatoids much beyond the convex prisms and antiprisms. In fact, it turns out thatthis family contains orientable polyhedra of positive genus, as well as nonorientablepolyhedra.

The only source of which we are aware, in which prismatoids other than Archimedeanprisms and antiprisms are discussed in some detail, is Sections 114 and 140 of [2].Uniform prismatoids (that is, prisms and antiprisms with regular polygons as faces) havebeen discussed by several authors, as part of more inclusive investigations of uniformpolyhedra in general. Coxeteret al. [3] give a list of uniform prisms and antiprisms, aswell as references to the earlier literature. The enumeration of uniform polyhedra givenby Coxeteret al. was shown to be complete by Sopov [16] and Skilling [15]. Har’El[9] gives skeletal illustrations of all nonprismatic uniform polyhedra, but illustratesonly pentagonal uniform prisms and antiprisms. In the more general case considered byBruckner [2] some of the unusual nonuniform isogonal prisms and antiprisms can beglimpsed. However, Br¨uckner’s presentation is completelyad hoc, with no particularguiding ideas and no clear classification or description principles; moreover, it is veryincomplete and misses some of the most interesting polyhedra of the types it purports toenumerate. We shall show that it is possible to describe many additional prismatoids in a

Isogonal Prismatoids 15

satisfactory and natural manner, and illustrate how varied are the shapes that prismatoidscan have.

In order to do this, we have to face another curious shortcoming regarding polyhedra—namely, the fact that the literature contains no satisfactory methods or ideas for a clas-sification of polyhedra of the general kind with which we are concerned. We attempt togive usable definitions of polyhedra, and of ways of deciding whether two polyhedra are“of the same type,” and apply them to investigate isogonal prismatoids.

This paper is organized as follows. In Section 2 we give precise definitions of theconcepts we need. In Section 3 we present a complete classification of prisms andantiprisms (not necessarily convex or uniform), while Sections 4 and 5 discuss otherisogonal prismatoids. Additional comments are collected in Section 6.

2. Polygons and Polyhedra

There is no generally accepted terminology for polyhedra in which faces may self-intersect, or intersect each other in various ways. Below we give the definitions whichseem appropriate to the topic at hand; however, since the polyhedra are built up ofpolygons, we first supply and illustrate the corresponding definitions for planar polygons.The exposition here follows the one in [7], simplified as appropriate for the restrictedclasses of polygons and polyhedra under investigation. Specifically, we are concernedhere only with polygons and polyhedra that are calledunicursal in [7], and only withepipedalrealizations of such polyhedra.

An abstract polygonis a fixed simplecircuit C, that is, a system consisting of a finite,cyclically ordered setV of distinct elements, and the setE of distinct unordered pairsof adjacent elements ofV. The elements ofV are the “vertices” of the polygon, and thepairs represent the “edges” of the abstract polygon.

A geometric polygon(or polygonfor short) P is the image of an abstract polygonC under a mapϕ which associates with each element ofV a point (vertexof P) in aEuclidean planeE2, and with each pair fromE the line segment (edgeof P) having asendpoints the images of the elements ofV that constitute the pair. IfP hasn edges wecall it an n-gon. We note that different vertices ofV may be represented by the samepoint of the plane; this does not affect the incidences of the vertices ofP with its edges,although it entails the possibility of edges that have coinciding vertices and are thereforerepresented by single-point line segments. Also, edges may cross or overlap in variousways, or even coincide.

In this paper we are interested mainly in triangles, quadrangles, and isogonal poly-gons, the latter including, in particular, regular polygons. Since the literature on isogonalpolygons is meager, and that on regular polygons contains a considerable amount of mis-leading statements, we present some details concerning these concepts. In analogy to thedefinition for polyhedra, a polygon is calledisogonalprovided its isometric symmetriesact transitively on its vertices. A polygonP is calledregular if its isometric symmetriesact transitively on theflagsof P, where a flag is the pair consisting of an edge and one ofits endpoints. (Many other definitions of regular polygons, equivalent to the one givenhere, are possible.) Ifk = [n/2], then there arek different regularn-gons, denoted by{n/d}, whered = 1, 2, . . . , k. (Throughout, we consider as equal all polygons which

16 B. Grunbaum

Fig. 2. The different isogonaln-gons withn = 5, 7, 9, or 15, and their symbols. All isogonaln-gons withoddn are regular. The vertices are denoted by the labels 1, 2, . . . ,n. Several labels near a single point indicatethat these vertices (although different as vertices of the polygon) are all represented by one point.

can be mapped onto each other by similarity transformations.) We note that, contrary tofrequently encountered assertions, a regularn-gon exists, and is a well-defined geometricobject, even if the integersn andd are not relatively prime; see Fig. 2 for examples.(Concerning this topic and its history see [5] and [7].)

Isogonal polygons seem to have been first investigated by Hess [10]; however, due tothe lack of a consistent point of view and disregard of the deeper insights of Meister [11]


Fig. 3. The different types of isogonal quadrangles. The types denoted 4-2 and 4-4 occur in continuousfamilies, whose shape varies from the square 4-1 to the quadrangle 4-3 in which two opposite sides arerepresented by segments of length 0, and the other two are represented by the same segment, and from thatquadrangle to the quadrangle 4-5 which has four sides represented by the same segment. The quadrangles 4-1and 4-5 are the only regular quadrangles.

and Wiener [19], this long work is unusable for any further investigations. A systematicapproach to isogonal polygons, including a full description and classification, appearsin [6]; here we only briefly recall the results. In order to avoid trivialities, we excludefrom further consideration isogonal polygons where all vertices coincide.

For oddn = 2k+1, the only isogonal polygons are the regular polygons{n/d}whered = 1, 2, . . . , k. In Fig. 2 we show all the types of isogonaln-gons forn = 5, 7, 9, and15; these examples are typical and should be sufficient to show what happens ifn anddare not relatively prime.

For evenn several cases need to be distinguished. A schematic illustration of thesomewhat special casen = 4 is shown in Fig. 3. For evenn ≥ 6, the isogonaln-gonsform [(n + 2)/4] families, of which [n/4] are continuous and can be parametrized byone real parameter each; an exceptional family occurs forn = 4k + 2, and consistsof the single (regular) polygon{n/(2k + 1)}. If n = 4k + 2 with k ≥ 1, then, foreachd = 1, 2, . . . , k, there is a continuous family denotedn/d of isogonaln-gons thatstarts with the regular polygon{n/d} and ends with the regular polygon{n/e}, wheree= 2k + 1− d. The casesn = 6 andn = 10 are illustrated in Figs. 4 and 5; the threecontinuous families that occur forn = 14 are shown in [6]. Similarly, forn = 4k withk ≥ 2 there arek continuous familiesn/d, whered = 1, 2, . . . , k; each familyn/dstarts at the regular polygon{n/d}, and—except for the familyn/k—ends at the regularpolygon{n/e}, wheree= 2k− d. The familyn/k reaches from{n/k} to {n/(2k)}. Thecasen = 8 is illustrated in Fig. 6.

Before starting the detailed discussion of isogonal prismatoids we have to makeexplicit what we understand as polyhedra, and how we distinguish between “types” ofpolyhedra. This definition turns out to be applicable to polygons as well, and to be anextension of the classification of polygons proposed long ago by Steinitz [17]. We beginby definingabstract polyhedra.

A finite family of abstract polygons is anabstract polyhedronprovided:

(i) Each “edge” of each of the polygons (which are the “faces” of the abstractpolyhedron) is an “edge” of precisely one other “face.”

(ii) All “faces” that contain a “vertex” form a single simple circuit.

18 B. Grunbaum

Fig. 4. The different types of isogonal hexagons. The polygons of types 6/1-1 to 6/1-7 form one continuousfamily, which starts at the regular polygon{6/1} and ends at the regular polygon{6/2}. The types denoted6/1-2, 6/1-4, and 6/1-6 depend on a real-valued parameter, and their shape varies between the shapes of thehexagons adjacent to them in the diagram. The regular hexagon{6/3} is isolated.

(iii) The family of “faces” is connected in the sense that any two belong to a chainof “faces” in which adjacent “faces” share an “edge.”

Since each “face” of an abstract polyhedron can be considered as the boundary of atopological disk, an abstract polyhedron can be interpreted as a cell complex decompo-sition of a 2-manifold. Two abstract polyhedra areisomorphicif there is an incidence-preserving bijection between their “vertices,” their “edges,” and their “faces.” Anabstractprismatoidhas two disjoint “faces” which together comprise all “vertices”; it isisogonalif the group of automorphisms acts transitively on the set of “vertices.” For any abstractisogonal polyhedron one can define its vertex-symbol, the cyclic list of sizes of the poly-gons at one (hence every) vertex of the polyhedron; of the different possible symbols,the one lexicographically first is usually chosen. Abstract prisms and antiprisms havevertex-symbols(4.4.n) and(3.3.3.n), respectively, wheren indicates the number of sidesof the basis.

A geometric polyhedron, or polyhedron Pfor short, is a representation of an abstractpolyhedron (said to be theunderlying abstract polyhedronof P) in the Euclidean 3-spaceE3, such that “vertices” are represented by points, “edges” by segments, and “faces” by(planar) polygons, giving thevertices, edges, andfacesof the polyhedron.

We say that a polyhedron isacoptic(from the Greekκoπτω, to cut) if:

(i) All its faces are simple polygons, so that they can be unambiguously represented(or replaced) bysimply-connected polygonal regions.

(ii) The intersection of any two such regions consists of a union (possibly empty) ofvertices and edges of each.

Clearly, convex polyhedra are acoptic, but so are many nonconvex polyhedra. Acopticpolyhedra are the ones for which cardboard models give a faithful representation; such amodel is, in fact, an embedding in 3- space of the 2-manifold determined by the abstractpolyhedron.

A polyhedron is calledaploic(from the Greekαπλooσ , onefold, simple) if, wheneverX andY are two distinct “vertices,” distinct “edges,” or distinct “faces” of the underlying


Fig. 5. The two continuous families of isogonal decagons. The types with odd suffix contain a single polygon,the others contain a continuum of polygons.

polyhedron, then the affine hulls affX and affY of X andY are distinct. The analogousdefinition applies to polygons as well. Aploic polyhedra can be considered as that gen-eralization of acoptic polyhedra to polyhedra with self-intersections which is closest tothe “naive” understanding. However, it should be pointed out that the tradition whichattempts to present aploic polyhedra (such as the Kepler–Poinsot regular polyhedra) bycardboard models is in many cases misguided and misleading; the most instructive mod-

20 B. Grunbaum

Fig. 6. The two continuous families of isogonal octagons.

els of nonacoptic aploic polyhedra are the skeletal ones, or those cardboard models inwhich the “hidden” parts are included and made visible through appropriate openings.

Each of the two planes that appear in the definition of a prismatoid may contain one ormore faces, or may not contain any faces; for isogonal prismatoids the figures in the twoplanes must be congruent, and in each plane the symmetries of the polyhedron must acttransitively on the vertices contained in that plane. The polygons contained in each of thetwo planes are said to form thebasisof the prismatoid; if no polygon is a basis, we saythat the prismatoid isbasis-free. An obvious consequence of the isogonality conditionis that the bases of any prismatoid which is not basis-free must be congruentisogonalpolygonsor isogonal compoundsof polygons. (An “isogonal compound” of polygons isa collection of polygons such that the isometric symmetries of the plane act transitivelyon the vertices of the collection.)

Since all vertices of any isogonal polyhedron are cospherical, and all vertices of anyprismatoid lie on two parallel planes, it is immediate that any aploic isogonal prismatoidcan have only triangles and quadrangles as faces of its mantle, and that it is either basis-free or each basis consists of a single isogonal polygon. The prisms and antiprisms (thatis, realizations of abstract prisms or antiprisms) are examples of the latter possibility,and we present their classification in the next section.

Before we can proceed with the classification, we need to define what constitutes a“type.” While this is a staple in the theory of convex polyhedra, and even for more general


Fig. 7. Representatives of all the nontrivial geometric types of quadrangles. Only the first three are acoptic,and only the ones in the first row are aploic.

acoptic polyhedra the problem of definition is not severe, in the present situation thereseems to be no reasonable definition to be found in the literature. Here by “reasonable” ismeant a definition that would be applicable to polyhedra of some degree of generality, andwhich reflects some natural properties we may wish to see preserved among polyhedrawe assign to the same type—but which, at the same time, is finer than the combinatorialclassification which is given by the abstract polyhedra. (We note that we find it appropriatenot to distinguish between two isogonal prismatoids—or more general polyhedra—ifone can be mapped onto the other by a nonsingular affinity that is compatible with alltheir isometric symmetries.) It is not clear whether our definition is reasonable for verygeneral polyhedra, but its application to aploic isogonal prismatoids appears to be bothconvenient and reasonable. The definition is analogous to the classification of polygonsthat goes back to Steinitz [17], and is similar to those in [12] and [7]; it is illustrated inFigs. 7–9.

Two polyhedraP0 andP1 are of thesame geometric typeprovided there is a continuousfamily P(t), 0 ≤ t ≤ 1, of polyhedra, whereP(0) = P0 and P(1) is P1 or a mirrorimage ofP1, and such that:

(i) All members of the family have the same underlying abstract polyhedronP.(ii) All members of the family have same group of isometric symmetries.

(iii) For any two distinct facesF1 andF2 of the underlying abstract polyhedronP,the affine hull of the union of the images ofF1 andF2 has the same dimensionfor every memberP(t) of the family.

In particular, two aploic isogonal prismatoids are of thesame geometric typeprovidedthey have the same underlying abstract polyhedron and the same symmetries, and thereis a continuous family of polyhedra connecting them in such a way that each polyhedron

22 B. Grunbaum

Fig. 8. Representatives of all the types of acoptic pentagons in the finer classification mentioned in the text,in which all edges and vertices are considered. One representative of each geometric type is indicated by anasterisk. The dotted lines indicate collinearity.

in the family is an aploic isogonal prismatoid with the same properties. Since here we areinterested only in the geometric types of the prismatoids under discussion, we usuallysimplify the language and speak of theirtype.

A finer classification would result if condition (iii) were expanded to include any twofaces in the wider sense (that is, the set that includes the faces, edges, and vertices) ofthe polyhedronP.

The definition of geometric type can obviously be applied to polygons as well, theonly change being the replacement of “faces” in condition (iii) by “edges.” In order toillustrate the concept of geometric type, and of the finer classification mentioned above,we show in Figs. 7 and 8 the different geometric types of quadrangles and of acopticpentagons, and in Fig. 9 the different types of acoptic polyhedra with five vertices.

In the next section we apply the definitions given here to the simplest prismatoids—theprisms and antiprisms.


Fig. 9. Representatives of all the eleven geometric types of acoptic polyhedra with five vertices. A short dashacross an edge indicates that the two faces incident with the edge are coplanar; a∨ indicates that the dihedralangle at the edge is concave. The underlying abstract polyhedron of the first three is a four-sided pyramid, forthe other eight it is a three-sided bipyramid.

3. The Classification of Prisms and Antiprisms

We begin by considering antiprisms, that is, polyhedra with vertex-symbol(3.3.3.n),wheren ≥ 3. We note that, for a givenn, all abstract antiprisms with vertex-symbol(3.3.3.n) are isomorphic. In any realization of an abstract antiprism by an isogonalpolyhedronP, the bases ofP have to be congruent regular polygons. For each regularpolygon{n/d} there exists a continuous family of antiprisms with bases congruent to{n/d}; this family can be parametrized by a real-valued parameter. The parameter canbe chosen to measure the twist of one of the bases with respect to the other; the two finalantiprisms in each family (and only they) have reflective symmetry. In Figs. 10–12 thefamilies of antiprisms withn = 3, 4, and 5 are illustrated. Either one or both extreme

24 B. Grunbaum

Fig. 10. Representatives of the continuous family of three-sided antiprisms. The two antiprisms marked bysingle or double asterisks are not aploic. The antiprism marked by two asterisks was mentioned by Sch¨onhardt[14] in a different context; the antiprisms to the left of it are acoptic.

members of each such family has a representative with regular mantle faces, that is, is auniform polyhedron and appears in the enumeration of Coxeteret al. [3]. Bruckner [2]mentions such antiprisms in parts 3 and 4 of Section 140, while the existence of the otherpolyhedra in each family is mentioned briefly (and in very unclear and confused terms,without any details) in parts 7 and 8 of Br¨uckner’s Section 140. An antiprism with basis{n/d} can be aploic only ifn andd are relatively prime; this is satisfied for the antiprismsin Figs. 10–12. Moreover, even ifn andd are relatively prime, for certain values of thetwist parameter the antiprism fails to be aploic due to coplanarity of distinct mantlefaces. These cases are marked by asterisks in Figs. 10–12; the nonaploic polyhedra ineach family that are marked by two asterisks seem to be particularly interesting.

According to the above definition, we can say that each of the continuous families ofantiprisms contains five different geometric types of aploic polyhedra: the two extremepolyhedra have more symmetries than the other members of the family, and the twononaploic polyhedra partition the intermediate members of the family into three types.If aploic, both extreme polyhedra can be represented by uniform polyhedra if and onlyif d > n/3.

Turning to prisms, with vertex-symbols(4.4.n), n ≥ 3, we note that, again, for eachnthere is a single combinatorial type of abstract prisms. Concerning geometric realizationsby isogonal polyhedra, we find that for each isogonaln-gon withodd n there exist two

Fig. 11. Representatives of the continuous family of four-sided antiprisms with basis{4/1}. The “twistparameter” increases from left to right in the first row, from right to left in the second. The two antiprismsmarked by asterisks are not aploic.


Fig. 12. Representatives of the two continuous families of five-sided antiprisms, based on{5/1} and{5/2},respectively. The parameter increases from left to right in the first and third rows of each half, and from rightto left in the second rows. The antiprisms marked by asterisks are not aploic.

(and only two) different prisms with bases congruent to thisn-gon; as mentioned above,this has to be a regularn-gon {n/d}. For reasons which should be evident from theexamples in Fig. 13, we say that one of them hasparallel bases, the otherantiparallelbases. The former (which may be denoted by a “p” appended to the symbol of theirbases) have rectangular mantle faces, the later (denoted similarly by an “a”) have self-intersecting isogonal quadrangles as mantle faces. These prisms are aploic if and only

26 B. Grunbaum

Fig. 13. The different types of isogonal n-prisms forn = 3, 5, and 9. The parallel or antiparallel position ofthe bases is indicated by the suffix “p” or “a” attached to the symbol that characterizes then-gonal basis.

if n andd are relatively prime; ifd = 1, then{n/d}p is acoptic. Obviously, all prismswith parallel bases can be represented by regular-faced polyhedra.

For even nthe situation is more interesting, due to the greater variety of isogonalpolygons in this case, as well as to the greater flexibility possible for the choice ofpolygons that form the mantle. We consider here only the case in whichn andd arerelatively prime. Then the prism is aploic unless the basis is a polygon with coincidingvertices; that is, the prism is aploic if its basis is aploic. (Ifn andd have a common divisork > 1, the appearance of the prisms is the same as forn′ = n/k andd′ = d/k, with k-tuples of vertices situated at each vertex of the prism corresponding ton′ andd′; all suchprisms are nonaploic.) For evenn > 4, for every nonregular polygon of each familyn/dthere are four different prisms with this polygon as basis; each regular polygon{n/d} isthe basis for two distinct prisms. The most interesting aspect of the situation is that, foreach pair(n, d) with d < n/4, all the prisms whose bases are in the familyn/d form asingle continuum of prisms. The “space” of these prisms, which can be parametrized bya real-valued parameter, is in fact homeomorphic to a circle. This is illustrated forn = 6,d = 1 in Fig. 14, and forn = 8, d = 1 in Fig. 15; similar diagrams result for other


Fig. 14. The hexagonal prisms. The diagram illustrates the continuous family in which to each regularpolygon as basis correspond two prisms, and to each nonregular isogonal polygon as basis correspond fourprisms. The family can be followed continuously from left to right in the first and third rows, from right to leftin the second and fourth; the final polyhedron is identical with the starting one. The prisms that correspond tothe same base are aligned vertically; those having regular bases are repeated in two rows. The prisms markedby asterisks are not aploic.

pairs(n, d). In each case other thann = 4d, there are eight prisms which are the singlemembers of their geometric type (only two of them are aploic), and eight geometric typesthat consist of a continuum of distinct aploic polyhedra; whend = 1 two of the latterfamilies consist of acoptic prisms, which were considered also by Robertson and Carter[13] and Robertson [12]. Br¨uckner [2] discusses prisms in Section 114 and in parts 1, 2,5, and 6 of Section 140; the sketchy presentation completely misses antiparallel prisms,as well as those prisms which are represented in the third row of each of our Figs. 14and 15.

The casen = 4, d = 1 is somewhat special; there is a single continuous family ofprisms with quadrangular basis, as illustrated in Fig. 16.

4. Other Aploic Prismatoids with Bases

Despite the great variety of forms possible for prisms and antiprisms, their abundanceis negligible compared with other isogonal prismatoids—even if only aploic ones areconsidered. We start by discussing the prismatoids with a basis, restricting attention to

28 B. Grunbaum

Fig. 15. The octagonal prisms, presented in a manner analogous to the one in Fig. 14. The prisms in whichthe bases have coinciding vertices are not aploic.


Fig. 16. The family of prisms with quadrangular basis. The family consists of one cycle, illustrated by thefirst two rows, together with two segments attached at different points of the cycle. All are aploic except theones with some coinciding vertices.

those in which all mantle faces are quadrangles. Since we are interested in polyhedraother than prisms, the number of such quadrangular faces incident with each vertex mustbe at least three. It is most astonishing thatnoneof these seems to have been mentionedanywhere in the literature; hence it is clear that they have no accepted names.

For aploic isogonal prismatoids that can be described collectively by the vertex-symbol(4.4.4.n), an investigation of the possible structure of the underlying abstractpolyhedra leads to precisely three distinct combinatorial types; instead of a listing offaces in the general case, we describe and illustrate the three kinds in Fig. 17 forn = 8.However, before continuing, it seems appropriate to acknowledge that the illustrationsare hard to interpret, and to describe schematic diagrams useful both for understandingthe structure of the polyhedra under discussion, and in establishing facts about them.

The diagrams in question, which are illustrated in Fig. 18, take advantage of the fact

30 B. Grunbaum

Fig. 17. Representatives of three different isomorphism types of aploic isogonal prismatoids, each withthe vertex-symbol(4.4.4.n), where n = 4k. The cases withn = 8 are shown. The polyhedronin (a) has as faces: 1,8,7,6,5,4,3,2,1; 11,12,13,14,15,16,17,18,11; 2,3,13,12,2; 4,5,15,14,4; 6,7,17,16,6;8,1,11,18,8; 1,15,5,11,1; 2,12,6,16,2; 3,17,7,13,3; 4,14,8,18,4; 1,2,16,15,1; 3,4,18,17,3; 5,6,12,11,5;7,8,14,13,7. It is orientable, of genusγ = n/4 = k. The faces of the polyhedron in (b) are:1,8,7,6,5,4,3,2,1; 11,12,13,14,15,16,17,18,11; 2,3,13,12,2; 4,5,15,14,4; 6,7,17,16,6; 8,1,11,18,8; 1,16,6,11,1;2,15,5,12,2; 3,18,8,13,3; 4,17,7,14,4; 1,2,15,16,1; 3,4,17,18,3; 5,6,11,12,5; 7,8,13,14,7. The faces of (c) are:1,8,7,6,5,4,3,2,1; 11,12,13,14,15,16,17,18,11; 2,3,12,13,2; 4,5,14,15,4; 6,7,16,17,6; 8,1,18,11,8; 1,15,4,18,1;3,17,6,12,3; 5,11,8,14,4; 7,13,2,16,7; 1,2,16,15,1; 3,4,18,17,3; 5,6,12,11,5; 7,8,14,13,7. The mantle of each ofthe three polyhedra consists of three kinds of quadrangles; one quadrangle of each kind is emphasized in thediagrams. The polyhedra in (b) and (c) are nonorientable, with Euler characteristicχ = 2− n/2.


Fig. 18. Diagrams used to represent and discuss isogonal prismatoids, as explained in the text. The diagramsin (a) represent the six-sided prisms shown in the leftmost columns of Fig. 14; the diagrams in (b) correspondto the prismatoids in Fig. 17.

that for isogonal prismatoids it is enough to specify which are the faces incident with onevertex. For ease of description we restrict the discussion to the case in which the bases areregular polygons. (In fact, there is no generality of structure lost by this simplification.)We start by choosing then points that represent the vertices of one of the bases. Thenfor one of the vertices we indicate all the faces incident with it, using the followingconventions (loosely derived from the idea that we are looking at the prismatoid fromfar above the center):

(i) We indicate only the faces of the mantle.(ii) A horizontal edge is indicated by a solid line, while an edge connecting vertices

of the different bases is indicated by a dashed line; vertical edges are not shown.(iii) A quadrangle with a pair of vertical edges is indicated by drawing a single bold

line (either solid or dashed).(iv) A vertical self-intersecting quadrangle with a pair of horizontal edges is indicated

by a pair of parallel thin lines, one solid and one dashed.(v) For nonvertical faces all four edges are indicated with thin lines, solid or dashed

as appropriate.

With a little patience and some practice, these diagrams can be used with advantagefor discussing the polyhedra they represent. For example, the two prisms in the leftmostcolumn of Fig. 14 are represented by the diagrams in Fig. 18(a), while the prismatoids(4.4.4.8) of Fig. 17 are represented by the three diagrams in Fig. 18(b).

It is easily verified that these three kinds of polyhedra with vertex- symbols(4.4.4.n)exist whenevern > 4 is a multiple of 4. Polyhedra of the kind shown in Fig. 17(a) areorientable, the other two kinds are nonorientable.

32 B. Grunbaum

Fig. 19. The continuous family of isogonal prismatoids with vertex-symbol (4.4.4.8), that includes thepolyhedron in Fig. 17(a). It starts at upper left, and snakes back and forth in alternate rows. They all are aploic,except for those with some coinciding vertices.

The examples in Figs. 17 and 18 have as bases regular octagons, but this is just forease of visualization: any isogonaln-gon with n = 4k ≥ 8 can serve as the basis foreach of these three kinds of polyhedra, and again the polyhedra form continuous families.For the particular polyhedron of Fig. 17(a) this family is illustrated by the diagrams inFig. 19. Details about the exact nature of these families have not been investigated sofar.

After the relatively simple case of prismatoids with vertex-symbols(4.4.4.n) weturn to the more complicated polyhedra that have vertex-symbols(4.4.4.4.n). To beginwith, there are polyhedra of this kind that are analogous to some extent to the ones oftype (4.4.4.n)—namely, involving a variety of shapes of quadrangles, including self-intersecting ones. Three examples of such polyhedra are shown in Fig. 20 and describedin its caption; the corresponding diagrams are shown in Fig. 21; in this case, however,


Fig. 20. Aploic isogonal prismatoids of three isomorphism types; all have vertex-symbol(4.4.4.4.n), withn = 4k in (a) andn = 2k in (b) and (c). The casesn = 8 are shown. All polyhedra of these types are nonori-entable, withχ = 2 − n. The faces of (a) are: 1,8,7,6,5,4,3,2,1; 11,12,13,14,15,16,17,18,11; 2,3,13,12,2;4,5,15,14,4; 6,7,17,16,6; 8,1,11,18,8; 1,15,5,11,1; 2,12,6,16,2; 3,17,7,13,3; 4,14,8,18,4; 1,2,11,12,1;3,4,13,14,3; 5,6,15,16,5; 7,8,17,18,7; 1,12,6,15,1; 2,16,5,11,2; 3,14,8,17,3; 4,18,7,13,4. The faces of (b) are:1,8,7,6,5,4,3,2,1; 11,18,17,16,15,14,13,12,11; 1,2,15,16,1; 2,3,16,17,2; 3,4,17,18,3; 4,5,18,11,4; 5,6,11,12,5;6,7,12,13,6; 7,8,13,14,7; 8,1,14,15,8; 1,16,6,11,1; 2,15,5,12,2; 3,18,8,13,3; 4,17,7,14,4; 5,18,8,15,5;6,11,1,16,6; 7,12,2,17,7; 8,13,3,18,8. The faces of (c) are: 1,8,7,6,5,4,3,2,1; 11,18,17,16,15,14,13,12,11;1,2,11,12,1; 2,3,12,13,2; 3,4,13,14,3; 4,5,14,15,4; 5,6,15,16,5; 6,7,16,17,6; 7,8,17,18,7; 8,1,18,11,8;1,12,6,15,1; 2,13,7,16,2; 3,14,8,17,3; 4,15,1,18,4; 5,16,2,11,5; 6,17,3,12,6; 7,18,4,13,7; 8,11,5,14,8. The man-tle of each polyhedron is formed by quadrangles of several shapes; one quadrangle of each kind is emphasized.

34 B. Grunbaum

Fig. 21. Diagrams corresponding to the prismatoids with vertex-symbol (4.4.4.4.8) shown in Fig. 20.

these examples are only some of the possibilities. Again, the use of regular octagonsas bases is just for ease of visualization; any isogonal polygon could be used, and thepolyhedra form continuous families. These have not been investigated in any detail, but itshould be noted that if the basis is a nonregular isogonal polygon, the number of distinctquadrangles in the mantle may be larger than in Fig. 20.

Besides these “strange” polyhedra, there are two families of isogonal prismatoids withvertex-symbol(4.4.4.4.n) that have mantles consisting entirely of simple quadrangleswhich, in suitable representatives, can all be made congruent. Thus they are, in a sense,closest to acoptic polyhedra and in particular to prisms and antiprisms. Prismatoids inthe two families are denoted by symbols of the formP(t0, t1; n) andA(t0, t1; n).

In Fig. 22 we show some of the simplest representatives of the first family; thecorresponding diagrams are shown in Fig. 23. Since these polyhedra are just the smallestinstances of prismatoids with more than three quadrangles incident with each vertex, wedescribe them by symbols that are easily adaptable to more general situations. Moreover,to simplify the exposition, for the time being we restrict attention to the case in which thebasis is a regular polygon. Other possibilities are considered later. We start by observingthat each mantle face is a trapezoidT1; the parallel edges ofT1 belong to the differentbasis planes of the prismatoid—but only one of these edges is an edge of the basispolygon. All these trapezoids are congruent. In the symbolP(t0, t1; n) of a prismatoidof this kind,n is the number of vertices of each base,t0 is thespanof that edge ofT1

which is also an edge of the basis, andt1 is the span of each side-edge ofT1. Here (andin what follows) by “span” we mean across how many steps along the vertices that arein the basis does the edge in question reach; note that this refers to the vertices as theyare encountered along the circle on which they lie, and not necessarily along the edgesof the basis polygon. The two bases are aligned, and the side-edges ofT1, besides havingspant1, also reach from one basis to the other. The fourth edge ofT1 has spant0 + 2t1,and is a diameter of then-gonal basis. Thereforen = 2t0 + 4t1. Since polyhedra are(by definition) connected, the positive integerst0 andt1 must be relatively prime, andt0must be odd.

The construction of polyhedra in the second family, which are denoted by the symbolA(t0, t1; n), can be described as follows; again we consider here only the case in whicheach basis is a regularn-gon. We take these twon-gons in an antiprismatic position, andnumber all 2n vertices consecutively as they appear on an orthogonal projection. Themantle facesT1 are, as in the first family, trapezoids which share one of their paralleledges with a basis. That edge has spant0, which therefore must be an even positive integer,


Fig. 22. Examples of isogonal prismatoids of the combinatorial type(4.4.4.4.n), belonging to the family ofpolyhedra denoted by the symbolP(t0, t1; n), which is explained in the text. One of the congruent quadranglesthat form the mantle is shown by heavy solid lines, while the two bases are represented by the heavy dashedlines.

Fig. 23. Diagrams corresponding to the prismatoids with vertex-symbol (4.4.4.4.8) shown in Fig. 20. Allmantle faces incident with the leftmost vertex are indicated in each diagram.

36 B. Grunbaum

while each of the side-edges of the trapezoid has spant1, which is an odd positive integer.Here the relation between the parameters isn = t0+2t1. Figure 24 shows some examplesof these prismatoids; the diagrams of three of them are contained in Fig. 25.

The two families of polyhedra with vertex-symbols(4.4.4.4.n) are the starting mem-bers (fork = 1) of two “superfamilies” of isogonal prismatoidsP(t0, t1, . . . , tk; n) andA(t0, t1, . . . , tk; n), with vertex-symbols(44k.n), wherek is any positive integer, andn isdetermined in a suitable manner. (The “exponential” notation in the vertex-symbol is justan abridgement of the “product” notation.) The definition of these families is analogousto the one we have seen in casek = 1, except that now the “free” edge (of spant0+ 2t1)of the trapezoidT1, instead of being a diameter of the basis polygon, is one of the paralleledges of a second trapezoidT2, whose side-edges have spant2. If k > 2, the fourth edge(of spant0+2t1+2t2) of T2 is one of the parallel edges of a trapezoidT3, the side-edgesof which have spant3, and so on in a self-explanatory manner. Hence the mantle consistsof k kinds of trapezoids, each represented by 2n congruent copies. For prismatoids fromthe familyP(t0, t1, . . . , tk; n) the value ofn is determined byn = 2t0+4(t1+· · ·+ tk),while for prismatoids fromA(t0, t1, . . . , tk; n) we haven = t0 + 2(t1 + · · · + tk); thelabeling of the vertices is in both cases the same as fork = 1. As in the casek = 1, weneedt0 to be positive, and odd in theP-families, while even in theA-families. In bothcases the parameterst0, t1, . . . , tk cannot all have a common factor, and some additionalconditions need to be satisfied in order to avoid unwanted coincidences. The preciseconditions have not been determined, but it appears that if all parameters are positive itis sufficient to require that none equals the sum of any collection of the others. Examplesof such polyhedra in the casek = 2 are shown in Figs. 26 and 27.

In addition to the two families of isogonal prismatoids with vertex-symbols(4h.n),where h ≡ 0(mod 4), there are two analogous families for whichh ≡ 2(mod 4).The polyhedra in these families are denoted by symbolsP(t0, t1, . . . , tk, t∗; n) andA(t0, t1, . . . , tk, t∗; n); they have vertex-symbols(44k+2.n). The polyhedra of these typesare constructed in exactly the same way as indicated by those parts of their symbol whichcoincide with the symbols of the polyhedra discussed above. The only additional part,t∗, indicates that the fourth edge of the trapezoidTk is also an edge of a self-intersectingquadrangleT∗ which has a pair of parallel edges, the distance between the parallel edgesbeingt∗, and the nonparallel edges having spann/2 in the first family andn in the second.Heren = 2t0+4(t1+· · ·+tk)+2t∗ for polyhedra with the symbolP(t0, t1, . . . , tk, t∗; n),andn = t0+ 2(t1+ · · · + tk)+ t∗ for polyhedra with the symbolA(t0, t1, . . . , tk, t∗; n).Again various conditions, which have not been completely determined so far, have tobe satisfied by the parameters in order to avoid unwanted coincidences or disconnected“polyhedra.” Examples of polyhedra of these types fork = 1 are shown in Figs. 28and 29.

As stated above, the discussion of theP- and A-families was conducted assumingthe basis polygons to be regular. However, the symmetry of these polyhedra allowsvarious modifications of the bases and of the polyhedra themselves, without loss oftheir character as isogonal prismatoids. As is easily verified, in the case of polyhedraP(t0, t1, . . . , tk; n)or P(t0, t1, . . . , tk, t∗; n), any isogonal polygon can serve as the basis;in fact, these polyhedra give rise to continuous families such as the ones in Figs. 14–16and 19. This is illustrated in Fig. 30 for the polyhedronP(1, 1; 6). Analogously, the basesof polyhedraA(t0, t1, . . . , tk; n) can be twisted with respect to each other, and then each


Fig. 24. Examples of isogonal prismatoids denoted byP(t0, t1; n), with vertex-symbol(4.4.4.4.n). ThesymbolP(t0, t1; n) is explained in the text. One of the congruent quadrangles that form the mantle is shownby heavy solid lines, while the two bases are represented by the heavy dashed lines.

38 B. Grunbaum

Fig. 25. Diagrams corresponding to the first two prismatoids with vertex-symbol (4.4.4.4.8), and the lastone, shown in Fig. 24. The vertices of one of the basis polygons are indicated by hollow dots, those of theother by solid dots.

Fig. 26. Examples of isogonal prismatoidsP(t0, t1, t2; n) with vertex-symbol(48.n). The meaning of thesymbol P(t0, t1, t2; n) is explained in the text. One of the trapezoidsT1 is shown in heavy solid lines, whileoneT2 is shown shaded. The two basis polygons are shown in heavy dashed lines.


Fig. 27. Examples of isogonal prismatoidsA(t0, t1, t2; n) with vertex-symbol(48.n). The meaning of thesymbol A(t0, t1, t2; n) is explained in the text. One of the trapezoidsT1 is shown in heavy solid lines, whileoneT2 is shown shaded. The two basis polygons are shown in heavy dashed lines.

trapezoid can be replaced by two triangles; if the replacement is done systematically,isogonal prismatoids with vertex-symbol(36k.n), wherek ≥ 1, are obtained. Thesecan be considered as natural relatives of the traditional antiprisms. As an example, oneisogonal prismatoid with vertex-symbol(36.4), obtained from the polyhedronA(2, 1; 4)by a slight twist of one basis polygon and replacement of each trapezoid by two triangles,is illustrated in Fig. 31. Clearly, the same method can also be applied to all isogonalprismatoidsP(t0, t1, . . . , tk; n) with regular polygons as the basis.

5. Basis-free Aploic Prismatoids

There are several interesting families of basis-free aploic polyhedra, some of which wedescribe next. However, these are only a small part of the possible polyhedra of this kind.A systematic investigation would seem to be a challenging but rewarding task.

(i) The simplest (and most widely known) are thesphenoids, that is, isogonal tetra-hedra, examples of which are shown in Fig. 32. If we had included “digons” amongpolygons, sphenoids would be antiprisms with digonal bases. All sphenoids are isohedralas well (that is, their symmetry group acts transitively on the faces); in the terminology

40 B. Grunbaum

Fig. 28. Examples of isogonal prismatoids of typeP(t0, t1, t∗; n), with vertex-symbol(46.n). The meaningof the symbolP(t0, t1, t∗; n) is explained in the text. One of the trapezoidsT1 is shown in heavy solid lines,while oneT∗ is shown shaded. The two basis polygons are shown in heavy dashed lines.


Fig. 29. Examples of isogonal prismatoids of typeA(t0, t1, t∗; n), with vertex-symbol(46.n). The meaningof the symbolA(t0, t1, t∗; n) is explained in the text. One of the trapezoidsT1 is shown in heavy solid lines,while oneT∗ is shown shaded. The two basis polygons are shown in heavy dashed lines.

42 B. Grunbaum

Fig. 30. The family of polyhedra obtained fromP(1, 1; 6) by continuously changing the bases from theregular hexagon to other isogonal hexagons. All these isogonal prismatoids have the same underlying abstractpolyhedron with vertex-symbol (4.4.4.4.6).


Fig. 31. A member of the continuous family isogonal prismatoids obtained from the polyhedronA(2, 1; 4)in Fig. 24 by a slight twist of one basis polygon with respect to the other, with appropriate replacement ofthe skew quadrangle by two triangles. The polyhedra in this family are orientable and have vertex-symbols(36.n). For the prismatoid shown,n = 4 and the faces are: 1,7,5,3,1; 2,4,6,8,2; 1,3,4,1; 3,5,6,3; 5,7,8,5; 7,1,2,7;2,8,3,2; 4,2,5,4; 6,4,7,6; 8,6,1,8; 1,4,8,1; 3,6,2,3; 5,8,4,5; 7,2,6,7; 2,1,5,2; 4,3,7,4; 6,5,1,6; 8,7,3,8.

of Grunbaum [7] such polyhedra are called “noble.” The sphenoids and the Platonic(regular) polyhedra are the only noble polyhedra that are acoptic; below we describe ad-ditional noble polyhedra which are not acoptic. There are two geometric types of aploicsphenoids, distinguished by their symmetry group. The less symmetric sphenoids forma continuous family that depends on one real parameter.

(ii) A general method for the generation of basis-free isogonal prismatoids uses theBoolean sumof two or more suitable isogonal prismatoids with bases. The prismatoidshave to be chosen so that their bases coincide in pairs, and those pairs are deleted fromthe new polyhedron. With appropriate choices, the resulting isogonal prismatoids areaploic; in fact, in some special cases they are even acoptic. The method of Boolean sumsis an open-ended one, since the number of prismatoids with bases that can be involvedin the formation of a basis-free prismatoid can be arbitrarily large; hence there is no

Fig. 32. The continuous family of sphenoids. The extreme polyhedra have greater symmetry than those ofthe intermediate type. All except the last are acoptic; the last one is not aploic, since all four faces are in thesame plane.

44 B. Grunbaum

Fig. 33. Examples of basis-free isogonal prismatoids obtained from (a) two antiprisms; (b) an antiprism anda prism; (c) two prisms. In each case the two starting polyhedra (shown at the left and in the center) must havecoinciding bases; the resulting polyhedron is shown at the right. The basis-free prismatoids in (b) and (c) areaploic, the one in (a) is not.

hope of giving a complete enumeration. Some of the possibilities are discussed in thefollowing paragraphs.

The simplest case is the Boolean sum of two antiprisms, or two prisms, or one prismsand one antiprism, which share both bases; eliminating these bases gives a basis-freeisogonal prismatoid. This is illustrated in Fig. 33. More complicated Boolean sums, eachinvolving four prisms, are shown by the examples in Fig. 34. The basis-free prismatoids


Fig. 34. Examples of basis-free aploic isogonal prismatoids (shown at the right) with vertex-symbol (4.4.4.4),obtained from four prisms each (shown to the left of the basis-free polyhedron). The basis-free prismatoid in(a) is a realization of the regular toroidal map with symbol{4, 4}4,0 in the notation of Coxeter and Moser [4].

in Figs. 33(b, c) and 34 are aploic, those in Fig. 34 are orientable. In fact, it is easyto verify that, understanding the faces of the prismatoid in Fig. 34(a) as determiningquadrangular regions, a realization of the regular toroidal map{4, 4}4,0 is obtained. Theresults of analogous Boolean sums involving four antiprisms are shown in Fig. 35. Thepolyhedron in Fig. 35(c) is not aploic, but it is noble; this type of noble polyhedra wasdescribed in [7] under the name “wreath polyhedra.”

Fig. 35. The polyhedra in (a) and (b) are examples of basis-free aploic isogonal prismatoids with vertex-symbol (3.3.3.3.3.3). Each is obtained from four antiprisms with bases coinciding in suitable pairs. Theunderlying abstract polyhedron is topologically a torus. The faces of the prismatoid in (a) are: 1,11,3,1;3,11,13,3; 3,13,5,3; 5,13,15,5; 5,15,1,5; 1,15,11,1; 1,3,12,1; 3,14,12,3; 3,5,14,3; 5,16,14,5; 5,1,16,5; 1,12,16,1;2,13,11,2; 2,4,13,2; 4,15,13,4; 4,6,15,4; 6,11,15,6; 6,2,11,6; 2;16,12,2; 2,12,4,2; 4,12,14,4; 4,14,6,4; 6,14,16,6;6,16,2,6. The faces in (b) are determined analogously. Suitable rotations of the upper bases yield families ofpolyhedra of the same type. Transitions between types lead to nonaploic polyhedra, some of which are isohedralas well; an example of such a polyhedron is shown in (c).

46 B. Grunbaum

Fig. 36. Examples of basis-free isogonal prismatoids obtained from two pentagonal antiprisms with commonbases. Each prismatoid is shown in skeletal as well as in surficial form—the latter being appropriate since thefaces are simple polygons (triangles) that do not cross each other. The middle one is acoptic; the other two,which represent the extremes of a continuous family of acoptic isogonal prismatoids, fail to be acoptic sincesome of their edges are in the planes of other faces.

As a particular cases of Boolean sums, if two suitable acoptic antiprisms are used,or an antiprism and a prism, acoptic isogonal tori can be obtained. Examples of suchtoroidal polyhedra, with vertex-symbol (3.3.3.3.3.3) are shown in Figs. 36 and 37. Likethe antiprisms or prisms used in their construction, these polyhedra depend on variousparameters; beyond certain parameter values they cease to be acoptic. Polyhedra of thiskind were described (along with isogonal acoptic polyhedra of higher genera) in [8].According to a private communication from Prof. J. M. Wills, some of these toroidalprismatoids were described earlier, by U. Brehm at a meeting in Oberwolfach in 1977;however, the only published account of this presentation [1, p. 438] contains no specifics,and does not mention isogonality.

Boolean sums of suitable prisms with prismatoids of typeP(t0, t1, . . . , tk; n) orP(t0, t1, . . . tk, t∗; n) can yield aploic basis-free isogonal prismatoids with vertex-symbol(42h), whereh ≥ 6. Other combinations using the same technique are possible as well; forexample, from antiprisms and prismatoids of typeA(t0, t1, . . . , tk; n) or A(t0, t1, . . . , tk,t∗; n), aploic basis-free isogonal prismatoids with vertex-symbols(3.3.3.42h), whereh ≥ 4, and many other polyhedra can be obtained.

(iii) Another remarkable family of basis-free isogonal prismatoids are thecrownpolyhedra; they were first described by Edmund Hess (see references in [7]) under thename “stephanoids” (from the Greek word for “crown”). Like the sphenoids, all crownpolyhedra are isohedral as well (thus they are noble). The crown polyhedra are of twokinds, which can be called the prismatic and the antiprismatic, each kind dependingon three positive integers as parameters; for appropriate values of these parameters,


Fig. 37. A continuous family (going from left to right in the upper two rows, and from right to left in the twobottom rows) of toroidal isogonal prismatoids, obtained from pairs of nine-sided antiprisms. Both skeletal andsurficial forms of each polyhedron are shown. All these polyhedra have combinatorial type (3.3.3.3.3.3). Exceptfor the starting and the ending members of the family, all polyhedra are acoptic. At two stages the polyhedracontain pairs of coplanar triangular faces, which can be replaced by rectangles; the resulting polyhedra havecombinatorial type (3.3.3.4.4) and can also be obtained directly from suitable prisms and antiprisms.

48 B. Grunbaum

Fig. 38. Two examples of crown polyhedra. The one in (a) is prismatic, while the one in (b) is antiprismatic.Crown polyhedra are not only isogonal but isohedral as well. One polygon in each polyhedron is emphasized.

the crown polyhedra are aploic. In all cases, the faces are congruent self-intersectingquadrangles. The two kinds are illustrated in Fig. 38 by one aploic representative each.For more details see Section 6 of [7].

6. Comments and Open Questions

(i) This paper developed from the observation that the presentation of the classification ofisogonal prisms and antiprisms in [2] is confused and incomplete. Since there appears tobe no other treatment of this topic, the author decided to write up his observations in whatbecame Section 3 of this note. However, further investigation showed that Brückner wasnot only deficient in the treatment of isogonal prisms and antiprisms, but that he missedcompletely the huge collection of isogonal prismatoids discussed in Sections 4 and 5above. (The only exception to this is his mention of crown polyhedra (“stephanoids”)which, curiously enough, he did not discuss in the presentation of isogonal polyhedra.)In the beginning we believed that Brückner (unconsciously?) wished to avoid includingin the discussion of isogonal polyhedra those that have self-intersecting quadranglesas faces, or polyhedra that do not have isohedral polar polyhedra; but the existence ofpolyhedra of typeP(t0, t1, . . . , tk; n), which he also failed to mention, disproves this idea.Now we are inclined to think that he simply did not look for any isohedra except those thatare isomorphic to uniform polyhedra. Why he would have thought this appropriate (inparticular, without mentioning it), and especially in view of the fact that he was awareof the existence of various noble polyhedra discovered by Hess, is really mystifying.However, on the other hand, it is equally hard to understand that during the almostfull century since the publication of Brückner’s book no attempt was made to rectify hisomission—in fact, to the best of the author’s knowledge, no mention of the shortcomingsof his enumeration of isogonal polyhedra made its way into print! Naturally, since hisenumeration of the isogonal prismatoids was so faulty, one has to wonder whether hisenumeration of the other isogonal polyhedra is complete. Since Brückner in this contextagain deferred the discussion of some noble polyhedra to a later section, and since his listof noble polyhedra is incomplete, a negative answer is obvious. A thorough investigationof these questions would appear to be long overdue.


Fig. 39. Examples of nonaploic prismatoids obtained by replacements of the bases of hexagonal or octagonalprisms. In the illustrations the mantle faces are the rectangles 1,2,12,11,1, etc. The upper basis in (a) is formedby the quadrangles 1,2,5,4,1, 3,4,1,6,3, and 5,6,3,2,5; by the hexagons 1,4,3,6,5,2,1 and 2,5,4,1,6,3,2 in (b),and by the quadrangles 1,4,5,8,1 and 3,6,7,2,3 and the octagon 1,2,7,8,5,6,3,4,1 in (c). The prismatoid (a) isnot orientable, while those in (b) and (c) are orientable; the polygons in the bases in (a) are congruent, while in(b) the two polygons in each basis have distinct shapes and in (c) they are even of different numbers of sides.

(ii) Nonaploic isogonal prismatoids arise not only as limiting cases of aploic families,but in many other ways as well. One general method, which is essentially again Booleanaddition, is to replace one polygonP by an edge-sharing family of polygons that havefree edges coinciding with those ofP, and that have, as a family, the same symmetry asP. This is illustrated by the examples in Fig. 39. Another application of the same idea isthe observation that given a rectangle and the two self-intersecting quadrangles whichhave the same vertices, any one of these three polygons can be replaced by the familyconsisting of the other two, or can be used to replace that family. For example, if thepolyhedron in Fig. 17(c) is combined in the manner discussed in (ii) of Section 5 with anoctagonal prism (with the deletion of the bases of both), the resulting polyhedron is notaploic since four faces of the original polyhedron have the same vertices as four of theeight mantle faces of the prism. However, if each such pair is replaced by the other self-intersecting quadrangle with the same vertices, the aploic isogonal prismatoid shownin Fig. 40 is obtained; this polyhedron has vertex-symbol (4.4.4.4), and it is orientablewith genus 0. Many other such examples can be formed, resulting in either aploic ornonaploic polyhedra.

50 B. Grunbaum

Fig. 40. An aploic isogonal prismatoid with vertex-symbol (4.4.4.4), derived from the polyhedron in Fig. 17(c)by a combination of the procedures described in (ii) of Sections 5 and 6. It is orientable, and of genus 0. The facesof the polyhedron are: 1,2,16,15,1; 3,17,18,4,3; 5,6,12,11,5; 7,13,14,8,7; 1,11,12,2,1; 3,4,14,13,3; 5,15,16,6,5;7,8,18,17,7; 1,15,4,18,1; 3,12,6,17,3; 5,11,8,14,5; 7,16,2,13,7; 1,18,8,11,1; 3,13,2,12,3; 5,14,4,15,4;7,17,6,16,7.

(iii) A complete determination of isogonal prismatoids with vertex-symbols(4.4.4.4.n), n ≥ 5, would be desirable, although probably quite hard. On the otherhand, even for small values ofn there are other interesting questions that may be pur-sued. For example, it is easy to verify that the prismatoidsA(2, 3; 8) and A(6, 1; 8)shown in Fig. 24 are isomorphic (have the same underlying abstract polyhedron). Thesame relation exists betweenA(2, 3,−1∗; 7) and A(4,−1, 5∗; 7), as well as betweenP(1, 2; 10) andP(3, 1; 10). This is clearly a consequence of some relations in the mod-ular arithmetic; however, the general behavior of the various prismatoids with respectto isomorphism has not been clarified so far. For example, there is a bijection betweenthe vertices ofA(2, 1, 3∗; 7) andA(4, 1, 1∗; 7) which maps faces to faces, but is not anisomorphism. Also open are questions regarding the character of the continuous familiesthat can be derived from the variousP- andA-prismatoids.

References

1. U. Brehm and W. K¨uhnel, Smooth approximation of polyhedral surfaces regarding curvatures.Geom.Dedicata12 (1982), 435–461.

2. M. Bruckner,Vielecke und Vielflache. Teubner, Leipzig, 1900.3. H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform polyhedra.Philos. Trans. Roy. Soc.

London Ser. A 246(1953/54), 401–450.4. H. S. M. Coxeter and W. O. J. Moser,Generators and Relations for Discrete Groups, 4th edn. Springer-

Verlag, Berlin, 1980.5. B. Grunbaum, Regular polyhedra.Companion Encyclopedia of the History and Philosophy of the Mathe-

matical Sciences, Vol. 2, I. Grattan-Guinness, ed. Routledge, London, 1994, pp. 866–876.6. B. Grunbaum, Metamorphoses of polygons.Proc.Strens Memorial Conference on Recreational Mathemat-


ics and Its History, R. K. Guy and R. E. Woodrow, eds. Mathematical Association of America, Washington,DC, 1994, pp. 35–48.

7. B. Grunbaum, Polyhedra with hollow faces.Proc. NATO–ASI Conference on Polytopes: Abstract, Convex,and Computational, Toronto, 1993, T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic’ Weiss, eds.Kluwer, Dordrecht, 1994, pp. 43–70.

8. B. Grunbaum and G. C. Shephard, Polyhedra with transitivity properties.Math. Rep. Acad. Sci. Canada6(1984), 61–66.

9. Z. Har’El, Uniform solutions for uniform polyhedra.Geom. Dedicata47 (1993), 57–110.10. E. Hess, Ueber gleicheckige und gleichkantige Polygone.Schriften Ges. Beford. gesammten Naturwiss.

Marburg, 10 (1874), 611–743 + plates.11. A. L. F. Meister, Generalia de genesi figurarum planarum et inde pendentibus earum affectionibus.Novi

Comm. Soc. Reg. Scient. Gotting. 1 (1769/70), 144–180 + plates.12. S. A. Robertson,Polytopes and Symmetry. London Mathematical Society Lecture Notes Series, Vol. 90.

Cambridge University Press, Cambridge, 1984.13. S. A. Robertson and S. Carter, On the Platonic and Archimedean solids.J. London Math. Soc. (2)2 (1970),

125–132.14. E. Sch¨onhardt,Uber die Zerlegung von Dreieckspolyedern in Tetraeder.Math. Ann. 98 (1927), 309–312.15. J. Skilling, The complete set of uniform polyhedra.Philos. Trans. Roy. Soc. London Ser. A 278 (1975),

111–135.16. S. P. Sopov, Proof of the completeness of the enumeration of uniform polyhedra.Ukrain. Geom. Sb. 8

(1970), 139–156.17. E. Steinitz, Polyeder und Raumeinteilungen.Encykl. math. Wiss. (Geom) 3 (3AB12), (1922), pp. 1–139.18. Webster’s Third New International Dictionary of the English Language. Encyclopædia Britannica, Chicago,

1966.19. C. Wiener,Uber Vielecke und Vielflache. Teubner, Leipzig, 1864.

Received April5, 1995.

Geometry - Springer · 18 B. Gr¨unbaum Fig. 4. The different types of isogonal hexagons. The...

Documents

Transcript of Geometry - Springer · 18 B. Gr¨unbaum Fig. 4. The different types of isogonal hexagons. The...